In this paper, we study weak amenability of Beurling algebras. To this end,
we introduce the notion inner quasi-additive functions and prove that for a
locally compact group G, the Banach algebra L1(G,Ο) is weakly
amenable if and only if every non-inner quasi-additive function in Lβ(G,1/Ο) is unbounded. This provides an answer to the question concerning
weak amenability of L1(G,Ο) and improve some known results in
connection with it